# Sanchez

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I really appreciate it when you take time to answer my questions. Thanks!

# 931 Actions

 Jan31 comment for any $a$ belongs to $\mathbb{C}$ $f(z_i)=a$ holds for $z_1,z_2,..;z_n$ Show that $\infty$ is not an essential singularity for $f$ using Picard. This will force $f$ to be a polynomial. Degree $n$ comes from your condition. Jan31 comment Open Subgroups of Affine algebraic groups You say it's connected - then $H$ should be the same as $G$, since the union of all other cosets of $H$ form an open set disjoint from $H$. Jan31 answered Inequality. $\frac{x^3}{y^2+z^2}+\frac{y^3}{z^2+x^2}+\frac{z^3}{x^2+y^2} \geq \dfrac{3}{2}.$ Jan31 awarded Organizer Jan31 revised A reference to study p-admissible functions edited tags Jan31 answered Exterior Algebra as quotient Jan30 comment A Dirichlet Convolution involving $\mu(n)$ and $\log n$ $\sum_{d|n} \mu(d)/d = \prod_{p|n} (1-1/p) = \phi(n)/n$ instead of $\phi(n)$. A similar mistake happens in your treatment of the second term - I believe that the final answer shouldn't have the $n$ factor in front. Jan29 comment $a^m+k=b^n$ Finite or infinite solutions? This probably follows from Siegel's theorem directly. Jan29 comment Primes of the form $n^{3} + 2$ It is however known that there are infinitely many primes of the form $x^3+2y^3$ (Heath-Brown). Jan29 comment Defining the Riemann-Roch space of a divisor @porkramen, write down such a linear combination, and let $Gal(\bar{K}/K)$ acts on it. All the $K(E)$ elements are fixed, so you would get a different linear combination unless all the coefficients are in $K$ already. If there is a different linear combination, take the difference with the original combination, we get linear dependence between the basis elements, contradiction. Jan29 comment Formal identity for sum of polynomials over a finite field. I meant to say that $\sigma(fg) = \sigma(f)\sigma(g)$ if $f$,$g$ are relatively prime. This, together with $F[x]$ is UFD, gives that LHS = $\prod_f \sum_{n=o}^{\infty} \sigma (f^n) |f|^{-ns}$. Simplifying this would probably give you what you want, but the answer below is definitely simpler. Jan29 comment Formal identity for sum of polynomials over a finite field. Have you used LHS is multiplicative? Jan29 comment How show that $\lim_{\varepsilon \rightarrow 0}\int_A h_\varepsilon(x)dx =0$, whenever $\bigg|\int_I h\bigg|\leq |I|^{1/2}$? @JacobSchlather, it gives you enough control indeed. First pick $\eta$, this will fix the number of intervals $I_k$ too, then choose a small enough $\epsilon$. Jan29 comment Defining the Riemann-Roch space of a divisor $\bar{K}$-basis, but the basis elements lie in $K(E)$. You can then show that if an element in $K(E)$ is written as linear sum of this basis, then the coefficients all lie in $K$. Jan29 answered Tangent Space Exercise Jan29 comment Finding a bound on $f'$ Fix an upper bound $B$ of $f$ on the unit circle. Use the given condition to get a bound of $f$ on a circle of radius $r$ (something like $Br$). Now use Cauchy's integral formula for $f'(z)$, and try to show that $f'(z)$ is bounded. Jan28 comment Computation of limit: $\lim\limits_{n \to \infty} \frac{1}{n}\left ( \cos1+\cos(\frac{1}{2})+\cos(\frac{1}{3}) +…+\cos (\frac{1}{n}) \right )$ @rlgordonma, wow you are quick. I thought I deleted that comment within 15s I posted it. Jan28 comment Inequality: $(a^3+a+1)(b^3+b+1)(c^3+c+1) \leq 27$ How is this different from the answer with 7 votes already? Jan28 comment Prerequisites for studying Hodge theory and the Hodge conjecture Someone who's well-versed in the area would write a nice answer, so let me just mention Voisin's books, "Hodge Theory and Complex Algebraic Geometry". Jan28 comment Resource for Vieta root jumping @CalvinLin, it seems hard to motivate an Olympiad trick, which doesn't even show up that much. (Appearance in IMO 2007 already attracted criticism) Not sure if higher degree polynomial can enter the picture.